Solve for x
x\in \left(\frac{3}{4},1\right)
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4x-3>0 x-1<0
For the quotient to be negative, 4x-3 and x-1 have to be of the opposite signs. Consider the case when 4x-3 is positive and x-1 is negative.
x\in \left(\frac{3}{4},1\right)
The solution satisfying both inequalities is x\in \left(\frac{3}{4},1\right).
x-1>0 4x-3<0
Consider the case when x-1 is positive and 4x-3 is negative.
x\in \emptyset
This is false for any x.
x\in \left(\frac{3}{4},1\right)
The final solution is the union of the obtained solutions.
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